Problem: $ 0.\overline{24} \div 0.\overline{3} = {?} $
First convert the repeating decimals to fractions. $\begin{align*} 100x &= 24.2424...\\ x &= 0.2424...\end{align*} $ $\begin{align*} 99x &= 24 \\ x &= \dfrac{24}{99}\end{align*} $ $\begin{align*} 10y &= 3.3333...\\ y &= 0.3333...\end{align*} $ $\begin{align*} 9y &= 3 \\ y &= \dfrac{3}{9}\end{align*} $ So, the problem becomes: $ \dfrac{24}{99} \div \dfrac{3}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{24}{99} \times \dfrac{9}{3} = {?} $ $ \phantom{\dfrac{24}{99} \times \dfrac{3}{9}} = \dfrac{24 \times 9}{99 \times 3} $ $ \phantom{\dfrac{24}{99} \times \dfrac{3}{9}} = \dfrac{24 \times \cancel{9}} {\cancel{99}11 \times 3} $ $ \phantom{\dfrac{24}{99} \times \dfrac{3}{9}} = \dfrac{24}{33} $ Simplify: ${= \dfrac{8}{11}}$